 |
 |
[3] viXra:1703.0273 [pdf] submitted on 2017-03-28 19:09:13
Authors: Arman Maesumi
Comments: 5 Pages.
Given a triangle ABC, the average area of an inscribed triangle RST whose vertices are uniformly distributed on BC, CA and AB, is proven to be one-fourth of the area of ABC. The average of the square of the area of RST is shown to be one-twelfth of the square of the area of ABC, and the average of the cube of the ratio of the areas is 5/144. A Monte Carlo simulation confirms the theoretical results, as well as a Maxima program which computes the exact averages.
Category: Geometry
[2] viXra:1703.0267 [pdf] submitted on 2017-03-28 08:30:43
Authors: Jan Hakenberg, Ulrich Reif
Comments: 5 Pages.
The derivation of multilinear forms used to compute the moments of sets bounded by subdivision surfaces requires solving a number of systems of linear equations. As the support of the subdivision mask or the degree of the moment grows, the corresponding linear system becomes intractable to construct, let alone to solve by Gaussian elimination. In the paper, we argue that the power iteration and the geometric series are feasible methods to approximate the multilinear forms. The tensor iterations investigated in this work are shown to converge at favorable rates, achieve arbitrary numerical accuracy, and have a small memory footprint. In particular, our approach makes it possible to compute the volume, centroid, and inertia of spatial domains bounded by Catmull-Clark and Loop subdivision surfaces.
Category: Geometry
[1] viXra:1703.0080 [pdf] replaced on 2021-08-19 15:05:47
Authors: Gaurav Biraris
Comments: 29 Pages.
The paper proposes generalization of geometric notion of vectors concerning dimensionality of the configuration space. Trivial mapping between an algebraic vector space and Euclidean space is possible as the Euclidean space is able to configure all elements of the algebraic vector space. Such configuration relies on the notion of globally valid directions those satisfy the vector axioms upon their direct product with lengths. We prove that, certain type of ordered direction exists in each number of Euclidean dimensions along which elements of vector spaces can be interpreted. We show that such general ordered directions equivalently exist at each point in Euclidean space and there exists a special metric for each kind of the ordered direction. An algebraic structure of addition and scaling exists for the direct product of such directions and path lengths along such directions. The path length is in terms of the special metric that comes with each dimension. We further show that this consideration satisfies the vector axioms and leads to the complete normed space within the Euclidean space. A mathematical framework is built with 3 lemmas, 8 theorems and a conjecture. Application of the framework to locally 3+1 dimensional universe leads to four fundamental versions as which a vector can exist geometrically. Thus any physical quantity in the universe should come in four versions of vectors as long as the underlying structure of spheres exists for the ordered directions.
Category: Geometry
| Third-Party Links: |
|